# Improving the Performance of Hydrological Model Parameter Uncertainty Analysis Using a Constrained Multi-Objective Intelligent Optimization Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. XAJ Model

#### 2.1.1. The Structure of the XAJ Model

#### 2.1.2. Parameters of the XAJ Model

#### 2.2. Data Sources

^{2}and encompasses 10 rainfall gauges, including Chengcun, Wangcun, Zuolong, and Tianli, etc. Chengcun station serves as the basin’s outlet hydrological station.

#### 2.3. Framework for Constrained Multi-Objective Intelligent Optimization Algorithm

## 3. Parameter Optimization in Consideration of Constraints

#### 3.1. Relationship between Soil Moisture and Model Parameters

#### 3.2. Analysis of the Flow Concentration Parameters

#### 3.3. Construction of the Penalty Function for Constraints

- The penalty function ensures that the soil moisture is always non-negative

_{max}represents the upper limit of parameter WM; and WM

_{min}represents the lower limit of the parameter WM. The remaining parameters are explained in the preceding sections.

- 2.
- The penalty function guarantees the constraint of CG ≥ CI ≥ CS

_{max}and CI

_{min}represent the upper and lower limits of the parameter CI, respectively; CG

_{min}represents the lower limit of the parameter CG, and CS

_{max}represents the upper limit of the parameter CS.

#### 3.4. Comparison of Parameter Optimization Results

#### 3.4.1. Analysis of Optimization Results

#### 3.4.2. Analysis of the Stability of the Algorithm

#### 3.4.3. Analysis of Runoff Relative Error

## 4. Sample Generation Based on Constrained Multi-Objective Intelligent Optimization

#### 4.1. Sample Generation Based on Single-Objective/Multi-Objective Optimization Algorithms

#### 4.1.1. Improved SCE-UA Algorithm

_{max}= 5 × 10

^{6}); the maximum number of allowable objective function improvement failures (k

_{max}= 50); the minimum objective function improvement rate (TOL

_{a}= 0.001%); and the interval of parameter convergence (TOL

_{λ}= 0.001%).

_{obs,t}represents the measured flow value at time t; Q

_{sim,t}represents the simulated flow at time t; Q

_{obs,mean}represents the mean value of the measured flow. The closer the NSE value is to one, the better the simulation results. The penalty factor λ is set to 10,000 for all parameters, with NSE as the optimization objective.

#### 4.1.2. Differential Evolution Algorithm

_{max}= 5,000,000. To uphold the diversity of the parameter population throughout the optimization process, Equation (4) is employed as the mutation strategy.

_{G}is the mutation point; X

_{rj,G}are mutually unequal random points; and F is the scaling factor. NSE is used as the objective function for the DE optimization algorithm.

#### 4.1.3. The NSGA-II Algorithm with Constraints

^{2}) and the mean absolute error (MAE) are commonly used as objective functions in model parameter optimization. Their expressions are as follows:

- (1)
- Coefficient of determination (R
^{2})

- (2)
- Mean absolute error (MAE)

^{2}differ. The NSE ranges from −∞ to 1, with values closer to 1 indicating better optimization results. MAE ranges from 0 to +∞, with lower values indicating better optimization results. R

^{2}ranges from 0 to 1, with values closer to 1 indicating better optimization results.

^{2}are maximization metrics. Therefore, in this study, 1 − NSE, MAE, and 1 − R

^{2}are used as the objective functions. By doing so, all three objective functions follow the rule of “the closer to 0, the better the optimization result”, thus making them suitable for the NSGA-II algorithm.

^{6}iterations of the XAJ model. The penalty factor λ used in the optimization process was set to 10,000.

#### 4.2. Evaluation Indicators for Sample-Generation Algorithms

#### 4.2.1. Indicators for Assessing the Prediction Bounds

- (1)
- Containing ratio (CR)

- (2)
- Average bandwidth (AB)

_{i}is the bandwidth of the prediction bounds for the discharge at time i; Q

^{u}

_{sim,i}and Q

^{l}

_{sim,i}represent the lower and upper prediction bounds of discharge, respectively. For a specific confidence level α, it is optimal for the bandwidth of the prediction bounds to be as narrow as possible. This enables the capture of crucial information regarding modeling uncertainty, making it more pertinent and valuable in relation to the forecasting concerns of the respective catchments.

#### 4.2.2. The Performance Indicators for Different Parameter Populations

- (1)
- Mean and variance of the Nash–Sutcliffe efficiency coefficient

- (2)
- Mean Euclidean distance

#### 4.3. Results and Analysis of the Sample-Generation Algorithms

^{2}), as the optimization objectives are demonstrated in Figure 8c,d. As can be observed from the histogram, the multi-objective optimization algorithm displays a more intricate distribution pattern and a wider distribution range compared to the single-objective optimization algorithm. This result suggests that the sampled dataset is dispersed across multiple centers, and the multi-objective optimization approach is better able to simultaneously find multiple “good points” and explore the parameter space.

^{2}) as the optimization objectives is shown in Figure 8e. The histogram displays a complex distribution pattern, and in comparison to the double-objective NSGA-II method, the parameter sample set obtained using the triple-objective NSGA-II method exhibits a broader distribution range. This indicates that the triple-objective NSGA-II method has a stronger ability to explore the parameter space.

^{2}) as optimization objectives is shown in Figure 9. It demonstrates the effective convergence of the NSGA-II algorithm towards the Pareto front of the double-objective optimization problem.

^{2}) as optimization objectives is shown in Figure 10. As depicted in the figure, the set of parameter samples converges towards a three-dimensional Pareto front.

#### 4.4. Containing Ratio (CR) and Average Bandwidth (AB) of Different Sample-Generation Algorithms

#### 4.5. Performance Comparison of Different Sample Generation Methods

#### 4.5.1. Performance Indicators for Different Parameter Populations

#### 4.5.2. Comparison of Diversity

^{2}) combination, and the population with three objectives shows even better diversity than the population with two objectives.

## 5. Conclusions and Outlook

- (1)
- On the basis of the numerical experiments conducted, it was demonstrated that WM has a significant impact on the positivity and negativity of the soil moisture, when other variables are kept fixed. It was found in this study that increasing WM reduced the likelihood of negative soil moisture, while decreasing WM increased the possibility of negative soil moisture. Other parameters, such as C, also had an effect on the soil moisture, while the effect of parameter B was not obvious.
- (2)
- The constraint of “soil moisture always non-negative” was introduced as a penalty function in the parameter optimization process. The penalty function penalized parameter sets that led to negative soil moisture, and thus the hydrological simulations with the optimized parameters did not have the phenomenon of “soil moisture less than zero”, and therefore achieved a better water balance.
- (3)
- The physical meaning of the flow concentration parameters was incorporated into the parameter optimization process as a constraint by using a penalty function treatment. The simulation results showed that after incorporating the constraint, the physical meaning of the flow concentration parameters was maintained, and there was no significant negative impact on the accuracy of the simulation.
- (4)
- Compared with the single-objective sample generation method, the sample population generated using the multi-objective method had better spatial exploration capability, while a similar degree of representativeness was maintained. The multi-objective method is a more suitable sample generation method for hydrological model parameter uncertainty analysis.

- (1)
- It should be noted that the coverage and average width of the envelope of the NSGA-II-generated parameter populations should be further improved to fulfill the requirements of uncertainty analysis and ensemble flood forecasting, and there is still room for improvement with respect to this method in future studies.
- (2)
- In this study, the sampling approach was based on a global optimization algorithm that prioritized “exploration” over “exploitation”. One area of improvement for the current algorithm would be to enhance its fine-grain search capability. This could be achieved by considering the introduction of gradient descent methods. Gradient descent is a popular optimization technique that iteratively adjusts the parameters in the direction of the steepest descent of the objective function. By incorporating gradient descent methods into the algorithm, it is possible to achieve a more precise and stable exploration of the parameter space, leading to improved optimization results.
- (3)
- The process of sample generation can indeed be time consuming. Generating a large number of samples may require substantial time and computational resources. Future research could explore multi-thread CPU/CPU acceleration techniques for speeding up the calculation process of the multi-objective sampling algorithm used in XAJ model parameter optimization [32,43]. This area has received limited attention, and investigating acceleration methods could lead to improvements in computational efficiency and scalability. Such research would contribute to the field of hydrological modeling by providing faster and more efficient approaches for parameter optimization.
- (4)
- In future research, expanding the scope of uncertainty analysis in hydrological modeling and prediction would be a valuable objective. While this paper focuses on uncertainty in model parameters, considering the uncertainty in model inputs and the hydrological model itself is crucial for a comprehensive analysis [44,45,46]. By incorporating these sources of uncertainty, the accuracy of the study could be enhanced, and a more holistic understanding of uncertainty in hydrological modeling and prediction could be provided. This extension would contribute to advancing the field and improving the reliability of hydrological assessments and forecasts.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Spatial distribution map for parameters WM, B, and C (the upper left figure is the three-dimensional spatial distribution map, the rest of the figure is the two-dimensional projection of the three-dimensional spatial distribution). In the figure, the red dots represent the parameter groups where the soil water content is negative, while the blue dots correspond to the parameter groups where the soil water content is always non-negative.

**Figure 5.**Histogram of parameters in the case of negative soil moisture (from left to right: the parameters WM, C, B, respectively).

**Figure 8.**Histogram illustrating the distribution of optimization results obtained using the following sample-generation algorithms: (

**a**) using Improved SCE-UA; (

**b**) using DE; (

**c**) using NSGA-II with (1 − NSE)-MEA as the optimization objectives; (

**d**) using NSGA-II with (1 − NSE)-(1 − R

^{2}) as the optimization objectives; (

**e**) using NSGA-II with (1 − NSE)-MEA-(1 − R

^{2}) as the optimization objectives). The red line is a trendline.

**Figure 9.**Pareto front with double-objective optimization ((1 − NSE)-MEA on the left; (1 − NSE)-(1 − R

^{2}) on the right).

**Figure 11.**Simulated hydrographs based on model parameters optimized using different sample-generation algorithms: (

**a**) using Improved SCE-UA; (

**b**) using DE; (

**c**) using NSGA-II with (1 − NSE)-MEA as the optimization objectives; (

**d**) using NSGA-II with (1 − NSE)-(1 − R

^{2}) as the optimization objectives; (

**e**) using NSGA-II with (1 − NSE)-MEA-(1 − R

^{2}) as the optimization objectives.

Parameters | Physical Meaning | Value Range | Parameter Type |
---|---|---|---|

K | Conversion coefficient of the potential evapotranspiration | 0.1~2 | Runoff generation |

WM | Tension water storage capacity (mm) | 50~300 | |

WUM | Tension water storage capacity for upper soil layer (mm) | 5~60 | |

WLM | Tension water storage capacity for lower soil layer (mm) | 10~90 | |

C | Transpiration coefficient of the deep soil layer | 0.01~0.5 | |

B | Power of tension water storage capacity curve | 0.1~2 | |

IM | Proportion of impervious area within the entire watershed | 0.001~0.5 | |

SM | Free water storage capacity (mm) | 1~60 | Runoff separation |

EX | Power of free water storage capacity curve | 0.01~2 | |

KG | Groundwater outflow coefficient | 0.01~0.69 | |

KI | Interflow outflow coefficient | 0.01~0.69 | |

CG | Recession coefficient of the groundwater reservoir | 0~1 | Flow concentration |

CI | Recession coefficient of the interflow reservoir | 0~1 | |

CS | Recession coefficient of river network | 0~1 | |

L | Lag time of river network flow concentration (d) | 0~10 | |

X | Muskingum flow routing parameters | −0.5~0.5 |

Name of Sub-Basin | Chengcun | Wangcun | Zhangyuankou | Didian | Dongkengwu |
---|---|---|---|---|---|

area weight | 0.1214 | 0.1121 | 0.1053 | 0.1258 | 0.0463 |

Name of sub-basin | Yonggongcheng | Zuolong | Fengcun | Tianli | Dalian |

area weight | 0.0726 | 0.116 | 0.0871 | 0.0882 | 0.1253 |

Model Parameter | K | WM | WUM | WLM | C | B | IM | SM |
---|---|---|---|---|---|---|---|---|

Parameter values | 0.98 | 50~300 | 5~60 | 10~90 | 0.01~0.5 | 0.1~2 | 0.01 | 16 |

Model parameter | EX | KG | KI | CG | CI | CS | L | X |

Parameter values | 1.5 | 0.13 | 0.57 | 0.99 | 0.8 | 0.2 | 1 | - |

**Table 4.**Statistical results of the parameters optimized using the SCE-UA algorithm before introducing the penalty function for the XAJ model.

No. | K | WM | WUM | WLM | C | B | IM | SM | EX | KG | CG | CI | CS | L | NSE | REV/% | Result |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1.036 | 141.492 | 22.141 | 84.351 | 0.5 | 0.518 | 0.001 | 25.907 | 0.799 | 0.472 | 0.724 | 0.157 | 0 | 1 | 0.84 | 9.025 | −1 |

2 | 1.041 | 136.365 | 22.261 | 79.104 | 0.5 | 0.536 | 0.001 | 25.866 | 0.794 | 0.471 | 0.723 | 0.156 | 0 | 1 | 0.84 | 9.097 | −1 |

3 | 1.118 | 97.055 | 24.546 | 26.287 | 0.5 | 1.091 | 0.001 | 24.372 | 0.525 | 0.48 | 0.696 | 0.073 | 0 | 1 | 0.841 | 10.675 | −1 |

4 | 1.143 | 95.237 | 26.236 | 12.009 | 0.5 | 1.107 | 0.001 | 24.284 | 0.524 | 0.479 | 0.693 | 0.066 | 0 | 1 | 0.841 | 11.153 | −1 |

5 | 1.143 | 95.92 | 25.376 | 12.642 | 0.5 | 1.091 | 0.001 | 24.305 | 0.524 | 0.479 | 0.694 | 0.066 | 0 | 1 | 0.841 | 11.088 | −1 |

6 | 1.108 | 99.331 | 23.81 | 36.474 | 0.5 | 1.039 | 0.001 | 24.42 | 0.532 | 0.48 | 0.698 | 0.079 | 0 | 1 | 0.841 | 10.573 | −1 |

7 | 1.143 | 95.39 | 26.571 | 11.559 | 0.5 | 1.108 | 0.001 | 24.281 | 0.521 | 0.48 | 0.692 | 0.062 | 0 | 1 | 0.841 | 11.182 | −1 |

8 | 1.143 | 95.787 | 26.215 | 12.031 | 0.5 | 1.099 | 0.001 | 24.316 | 0.524 | 0.222 | 0.068 | 0.693 | 0 | 1 | 0.841 | 11.163 | −1 |

9 | 1.111 | 105.238 | 16.827 | 51.238 | 0.498 | 0.678 | 0.001 | 26.095 | 0.955 | 0.242 | 0.217 | 0.732 | 0 | 1 | 0.84 | 10.224 | −1 |

10 | 1.095 | 95.21 | 23.974 | 35.444 | 0.5 | 1.108 | 0.001 | 24.256 | 0.539 | 0.482 | 0.701 | 0.1 | 0.002 | 1 | 0.84 | 10.107 | −1 |

11 | 1.119 | 95.907 | 27.001 | 10.017 | 0.5 | 1.106 | 0.001 | 24.347 | 0.522 | 0.221 | 0.079 | 0.699 | 0 | 1 | 0.841 | 10.481 | −1 |

12 | 1.111 | 96.002 | 23.896 | 36.665 | 0.5 | 1.099 | 0.002 | 24.369 | 0.526 | 0.22 | 0.082 | 0.696 | 0 | 1 | 0.841 | 10.645 | −1 |

13 | 0.974 | 126.482 | 59.997 | 31.483 | 0.5 | 0.785 | 0.001 | 25.074 | 0.568 | 0.488 | 0.707 | 0.1 | 0 | 1 | 0.84 | 8.471 | −1 |

14 | 1.118 | 95.868 | 26.975 | 10.005 | 0.5 | 1.107 | 0.001 | 24.347 | 0.523 | 0.481 | 0.698 | 0.074 | 0 | 1 | 0.841 | 10.446 | −1 |

15 | 0.975 | 126.526 | 59.98 | 31.502 | 0.5 | 0.787 | 0.001 | 25.08 | 0.566 | 0.489 | 0.706 | 0.097 | 0 | 1 | 0.84 | 8.505 | −1 |

16 | 1.143 | 95.628 | 26.988 | 11.006 | 0.499 | 1.105 | 0.001 | 24.262 | 0.522 | 0.477 | 0.693 | 0.071 | 0 | 1 | 0.841 | 11.212 | −1 |

17 | 0.974 | 126.612 | 59.995 | 31.614 | 0.5 | 0.782 | 0.001 | 25.105 | 0.565 | 0.488 | 0.707 | 0.097 | 0 | 1 | 0.84 | 8.473 | −1 |

18 | 1.118 | 96.464 | 26.959 | 10.036 | 0.5 | 1.101 | 0.001 | 24.356 | 0.519 | 0.481 | 0.697 | 0.07 | 0 | 1 | 0.841 | 10.457 | −1 |

19 | 1.115 | 95.787 | 26.88 | 10.003 | 0.5 | 1.108 | 0.001 | 24.341 | 0.523 | 0.482 | 0.698 | 0.077 | 0 | 1 | 0.841 | 10.343 | −1 |

20 | 1.042 | 136.521 | 22.011 | 79.507 | 0.5 | 0.547 | 0.001 | 25.805 | 0.795 | 0.233 | 0.17 | 0.725 | 0 | 1 | 0.84 | 9.097 | −1 |

**Table 5.**Statistical results of the parameters optimized using the SCE-UA algorithm after introducing the penalty function for the XAJ model.

No. | K | WM | WUM | WLM | C | B | IM | SM | EX | KG | CG | CI | CS | L | NSE | REV/% | Result |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.978 | 151.458 | 43.44 | 73.018 | 0.484 | 0.499 | 0.001 | 26.057 | 0.808 | 0.477 | 0.725 | 0.174 | 0 | 1 | 0.84 | 8.3042 | 1 |

2 | 0.974 | 142.438 | 60 | 47.436 | 0.392 | 0.51 | 0.001 | 25.464 | 1.118 | 0.082 | 0.937 | 0.582 | 0 | 1 | 0.84 | 8.4069 | 1 |

3 | 1.033 | 145.024 | 22.084 | 87.938 | 0.5 | 0.509 | 0.001 | 25.911 | 0.806 | 0.469 | 0.724 | 0.165 | 0 | 1 | 0.84 | 8.9905 | 1 |

4 | 0.974 | 144.088 | 60 | 49.088 | 0.4 | 0.52 | 0.001 | 26.055 | 0.798 | 0.48 | 0.722 | 0.16 | 0 | 1 | 0.84 | 8.4601 | 1 |

5 | 0.974 | 143.098 | 60 | 48.096 | 0.395 | 0.527 | 0.001 | 25.982 | 0.771 | 0.478 | 0.723 | 0.154 | 0 | 1 | 0.84 | 8.4264 | 1 |

6 | 1.033 | 144.58 | 21.895 | 87.587 | 0.5 | 0.509 | 0.001 | 25.924 | 0.8 | 0.474 | 0.723 | 0.154 | 0 | 1 | 0.84 | 8.9652 | 1 |

7 | 0.974 | 144.666 | 59.998 | 49.667 | 0.404 | 0.516 | 0.001 | 26.093 | 0.787 | 0.478 | 0.724 | 0.157 | 0 | 1 | 0.84 | 8.4827 | 1 |

8 | 1.108 | 146.437 | 18.064 | 39.638 | 0.499 | 0.453 | 0.001 | 26.024 | 0.829 | 0.465 | 0.725 | 0.143 | 0 | 1 | 0.84 | 10.2454 | 1 |

9 | 0.974 | 145.15 | 60 | 48.983 | 0.409 | 0.515 | 0.001 | 26.071 | 0.8 | 0.477 | 0.724 | 0.164 | 0 | 1 | 0.84 | 8.4895 | 1 |

10 | 0.974 | 145.548 | 59.997 | 50.521 | 0.41 | 0.51 | 0.001 | 26.079 | 0.829 | 0.48 | 0.724 | 0.174 | 0 | 1 | 0.84 | 8.5164 | 1 |

11 | 0.974 | 144.76 | 60 | 49.76 | 0.405 | 0.515 | 0.001 | 26.073 | 0.799 | 0.479 | 0.723 | 0.161 | 0 | 1 | 0.84 | 8.4878 | 1 |

12 | 0.974 | 145.883 | 59.999 | 50.875 | 0.412 | 0.511 | 0.001 | 26.108 | 0.795 | 0.482 | 0.722 | 0.148 | 0 | 1 | 0.84 | 8.526 | 1 |

13 | 1.034 | 146.055 | 21.843 | 89.212 | 0.5 | 0.503 | 0.001 | 25.926 | 0.802 | 0.473 | 0.723 | 0.156 | 0 | 1 | 0.84 | 9.0291 | 1 |

14 | 1.031 | 144.301 | 21.998 | 87.3 | 0.5 | 0.511 | 0.001 | 25.797 | 0.816 | 0.455 | 0.732 | 0.204 | 0 | 1 | 0.84 | 8.9008 | 1 |

15 | 1.033 | 144.783 | 22.074 | 87.708 | 0.5 | 0.509 | 0.001 | 25.928 | 0.804 | 0.473 | 0.723 | 0.158 | 0 | 1 | 0.84 | 8.9856 | 1 |

16 | 0.974 | 145.049 | 50.047 | 35.002 | 0.406 | 0.515 | 0.001 | 26.06 | 0.811 | 0.473 | 0.728 | 0.178 | 0 | 1 | 0.84 | 7.7192 | 1 |

17 | 1.034 | 145.284 | 21.959 | 88.325 | 0.5 | 0.506 | 0.001 | 25.94 | 0.801 | 0.47 | 0.724 | 0.159 | 0 | 1 | 0.84 | 9.022 | 1 |

18 | 0.974 | 144.859 | 59.996 | 49.86 | 0.405 | 0.518 | 0.001 | 26.156 | 0.795 | 0.49 | 0.72 | 0.131 | 0 | 1 | 0.84 | 8.4864 | 1 |

19 | 1.108 | 147.171 | 18.299 | 33.741 | 0.499 | 0.449 | 0.001 | 26.079 | 0.823 | 0.466 | 0.726 | 0.138 | 0 | 1 | 0.84 | 10.1358 | 1 |

20 | 1.033 | 144.831 | 22.069 | 87.763 | 0.5 | 0.508 | 0.001 | 25.916 | 0.799 | 0.471 | 0.724 | 0.159 | 0 | 1 | 0.84 | 8.9874 | 1 |

Algorithm | Mean Value of NSE | Variance of NSE | Range of NSE |
---|---|---|---|

SCE-UA | 0.8406 | 0.0005 | 0.001 |

Constrained SCE-UA | 0.8400 | 0.0000 | 0.000 |

Algorithm | SCE-UA | Constrained SCE-UA |
---|---|---|

Mean value of Euclidean distance | 37.59 | 32.88 |

Variance in Euclidean distance | 755.49 | 531.173 |

Algorithm | Lower Edge | 25% | Median | Mean | 75% | Upper Edge |
---|---|---|---|---|---|---|

SCE-UA | 8.47 | 9.09 | 10.12 | 10.45 | 10.88 | 11.21 |

Constrained SCE-UA | 7.72 | 8.47 | 8.52 | 8.78 | 8.99 | 9.03 |

Year | Improved SCE-UA | DE | ||||
---|---|---|---|---|---|---|

CR | AB | NSE | CR | AB (×10^{−3}) | NSE | |

1986 | 0.01 | 0.04 | 0.850 | 0 | 1.85 | 0.854 |

1987 | 0.01 | 0.05 | 0.938 | 0 | 2.2 | 0.944 |

1988 | 0 | 0.05 | 0.880 | 0 | 1.71 | 0.873 |

1989 | 0.01 | 0.05 | 0.910 | 0 | 2.16 | 0.902 |

1990 | 0 | 0.04 | 0.795 | 0 | 1.84 | 0.803 |

1991 | 0 | 0.05 | 0.794 | 0 | 1.88 | 0.791 |

1992 | 0.01 | 0.04 | 0.784 | 0 | 1.79 | 0.785 |

1993 | 0.01 | 0.06 | 0.835 | 0.08 | 2.16 | 0.832 |

1994 | 0 | 0.04 | 0.805 | 0 | 1.49 | 0.812 |

1995 | 0 | 0.04 | 0.826 | 0 | 1.79 | 0.830 |

1996 | 0 | 0.05 | 0.829 | 0 | 1.9 | 0.822 |

1997 | 0 | 0.05 | 0.780 | 0 | 1.93 | 0.789 |

1998 | 0 | 0.05 | 0.740 | 0 | 1.87 | 0.741 |

1999 | 0.01 | 0.06 | 0.909 | 0 | 2.05 | 0.907 |

ALL | 0 | 0.05 | 0.840 | 0 | 1.90 | 0.841 |

Year | NSGA-II/(1 − NSE)-MEA | NSGA-II/(1 − NSE)-(1 − R^{2}) | NSGA-II/(1 − NSE)-MEA-(1 − R^{2}) | ||||||
---|---|---|---|---|---|---|---|---|---|

CR | AB | NSE | CR | AB | NSE | CR | AB | NSE | |

1986 | 0.2 | 1.96 | 0.841 | 0.03 | 0.57 | 0.852 | 0.2 | 2.55 | 0.843 |

1987 | 0.21 | 2.07 | 0.935 | 0.06 | 0.61 | 0.941 | 0.29 | 2.68 | 0.946 |

1988 | 0.35 | 2.01 | 0.878 | 0.03 | 0.47 | 0.878 | 0.38 | 2.43 | 0.874 |

1989 | 0.28 | 2.18 | 0.908 | 0.05 | 0.57 | 0.902 | 0.33 | 2.67 | 0.901 |

1990 | 0.27 | 1.85 | 0.803 | 0.05 | 0.55 | 0.786 | 0.28 | 2.25 | 0.800 |

1991 | 0.48 | 2.29 | 0.795 | 0.02 | 0.45 | 0.795 | 0.51 | 2.72 | 0.803 |

1992 | 0.39 | 1.86 | 0.775 | 0.06 | 0.53 | 0.777 | 0.45 | 2.43 | 0.784 |

1993 | 0.26 | 2.54 | 0.838 | 0.06 | 0.59 | 0.844 | 0.31 | 2.98 | 0.841 |

1994 | 0.33 | 1.87 | 0.813 | 0.04 | 0.37 | 0.804 | 0.37 | 2.26 | 0.813 |

1995 | 0.4 | 2.12 | 0.820 | 0.04 | 0.44 | 0.832 | 0.49 | 2.62 | 0.830 |

1996 | 0.33 | 2.38 | 0.831 | 0.03 | 0.57 | 0.837 | 0.37 | 2.94 | 0.838 |

1997 | 0.19 | 2.06 | 0.788 | 0.05 | 0.53 | 0.772 | 0.26 | 2.55 | 0.785 |

1998 | 0.41 | 2.41 | 0.749 | 0.02 | 0.37 | 0.744 | 0.45 | 2.92 | 0.745 |

1999 | 0.46 | 2.42 | 0.917 | 0.03 | 0.48 | 0.903 | 0.53 | 2.96 | 0.913 |

ALL | 0.32 | 2.15 | 0.841 | 0.04 | 0.51 | 0.840 | 0.37 | 2.64 | 0.841 |

Algorithms | Improved SCE | DE | NSGA-II | NSGA-II | NSGA-II |
---|---|---|---|---|---|

/(1 − NSE)-MEA | /(1 − NSE)-(1 − R^{2}) | /(1 − NSE)-MEA-(1 − R^{2}) | |||

Means | 0.8400 | 0.8410 | 0.8380 | 0.8390 | 0.8380 |

Variances | 5.6000 × 10^{−8} | 3.0110 × 10^{−6} | 0.0023 | 0.0005 | 0.0020 |

Algorithms | Improved SCE | DE | NSGA-II | NSGA-II | NSGA-II |
---|---|---|---|---|---|

((1 − NSE)-MEA) | (1 − NSE)-(1 − R^{2}) | ((1 − NSE)-MEA-(1 − R^{2})) | |||

Mean Euclidean distance | 0.3825 | 0.5795 | 11.6062 | 3.6339 | 33.5685 |

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**MDPI and ACS Style**

Liu, X.; Kan, G.; Ding, L.; He, X.; Liu, R.; Liang, K.
Improving the Performance of Hydrological Model Parameter Uncertainty Analysis Using a Constrained Multi-Objective Intelligent Optimization Algorithm. *Water* **2023**, *15*, 2700.
https://doi.org/10.3390/w15152700

**AMA Style**

Liu X, Kan G, Ding L, He X, Liu R, Liang K.
Improving the Performance of Hydrological Model Parameter Uncertainty Analysis Using a Constrained Multi-Objective Intelligent Optimization Algorithm. *Water*. 2023; 15(15):2700.
https://doi.org/10.3390/w15152700

**Chicago/Turabian Style**

Liu, Xichen, Guangyuan Kan, Liuqian Ding, Xiaoyan He, Ronghua Liu, and Ke Liang.
2023. "Improving the Performance of Hydrological Model Parameter Uncertainty Analysis Using a Constrained Multi-Objective Intelligent Optimization Algorithm" *Water* 15, no. 15: 2700.
https://doi.org/10.3390/w15152700